Mode-tuning sense interface

ABSTRACT

A MEMS capacitive sensing interface includes a sense capacitor having a first terminal and a second terminal, and having associated therewith a first electrostatic force. Further included in the MEMS capacitive sensing interface is a feedback capacitor having a third terminal and a fourth terminal, the feedback capacitor having associated therewith a second electrostatic force. The second and the fourth terminals are coupled to a common mass, and a net electrostatic force includes the first and second electrostatic forces acting on the common mass. Further, a capacitance measurement circuit measures the sense capacitance and couples the first terminal and the third terminal. The capacitance measurement circuit, the sense capacitor, and the feedback capacitor define a feedback loop that substantially eliminates dependence of the net electrostatic force on a position of the common mass.

RELATED APPLICATIONS

This patent application is a divisional application of U.S. patentapplication Ser. No. 13/720,984, filed on Dec. 19, 2012, by BuisCagdaser, et al., and entitled, “MODE-TUNNING SENSE INTERFACE”, thedisclosures of which are incorporated herein, in their entirety, byreference.

BACKGROUND

Capacitive position sensing is a common means of detecting displacementin MEMS transducers.

Mechanical parameters of a MEMS device generally determine essentialaspects of the transducer design such as sensitivity, noise performance,and MEMS dynamics (resonance frequency, quality factor, settling timeetc.). For instance, the mass and the spring of a MEMS device determinethe resonance frequency in accordance with the following relationship:

$\begin{matrix}{{f_{res} = \sqrt{\frac{k_{x}}{m}}},} & {{Eq}.\mspace{14mu} (1)}\end{matrix}$

Where k_(x) represents the spring, f_(res) represents the resonantfrequency and m represents the mass. The damping, b, determines theBrownian noise force as follows:

F _(B)=√{square root over (4k _(B) Tb)},  Eq. (2)

and all three determine the quality factor of the MEMS system, asfollows:

$\begin{matrix}{Q = {\frac{\sqrt{k_{x}m}}{b}.}} & {{Eq}.\mspace{14mu} (3)}\end{matrix}$

Often, MEMS dynamics are also influenced by factors other thanmechanical parameters. For example, a parallel-plate sense capacitor canundesirably introduce electrostatic spring softening. Such a device isoften unavoidable especially when MEMS motion is perpendicular to thedevice layer that the MEMS device is built in. The impact of springsoftening can be modeled by an additional spring (k_(v)) acting on themass. The electrostatic spring constant k_(v) is determined by thesecond derivative of the sense capacitance with respect to the positionand the high-voltage bias (V_(b)) as follows:

$\begin{matrix}{k_{v} = {{- \frac{1}{2}}\frac{^{2}C_{s}}{x^{2}}{V_{b}^{2}.}}} & {{Eq}.\mspace{14mu} (4)}\end{matrix}$

In the presence of spring softening, the resonance frequency of the MEMSsystem is determined by the sum of all springs acting on the mass, asfollows:

$\begin{matrix}{f_{res} = {\sqrt{\frac{k_{x} + k_{v}}{m}}.}} & {{Eq}.\mspace{14mu} (5)}\end{matrix}$

As the bias voltage increases or the parallel-plate gap decreases,spring softening can result in a net negative spring constant causinginstability also known as the “pull-in.” At smaller degrees, springsoftening can introduce significant variations to resonance frequencyand transducer sensitivity. Thus, stability of high-voltage bias and theparallel-plate sense capacitor's gap can become particularly importantsince they can introduce temperature and package dependence to thetransducer via spring softening.

In some prior art techniques, spring softening is avoided inparallel-plate actuators by maintaining a constant charge on theactuator capacitor. In a constant charge actuator, displacement iscontrolled by the amount of charge stored on the actuator capacitorinstead of the voltage difference applied across its terminals. Thedrive circuit controls the amount of charge flow to the actuatorcapacitor. Thus, the voltage across the actuator capacitor is free tofluctuate in response to actuator displacements. Such operation, alsoknown as the “charge control”, eliminates position dependence ofelectrostatic force, and, hence, the spring-softening in parallel-plateactuators.

In contrast to the charge controlled actuator, capacitive sensing oftenresults in charge transfer between the sense capacitor and thecapacitance measurement circuit. In a traditional trans-capacitanceimplementation, the capacitance measurement circuit uses a knowncapacitance C_(fb) to convert this charge reading into an outputvoltage. This scheme however, results in an unwanted position-dependentforce and introduces spring softening in the sense capacitor.

There is thus a need for a MEMS capacitive sensing interface withreduced electrostatic spring softening effect.

SUMMARY

Briefly, an embodiment of the invention includes a MEMS capacitivesensing interface that has a sense capacitor. The sense capacitor has afirst terminal and a second terminal and has associated therewith afirst electrostatic force. Further included in the MEMS capacitivesensing interface is a feedback capacitor having a third terminal and afourth terminal, the feedback capacitor having associated therewith asecond electrostatic force. The second and the fourth terminals arecoupled to a common mass, and a net electrostatic force includes thefirst and second electrostatic forces acting on the common mass.Further, a capacitance measurement circuit measures the sensecapacitance and couples the first terminal and the third terminal. Thecapacitance measurement circuit, the sense capacitor, and the feedbackcapacitor define a feedback loop that substantially eliminatesdependence of the net electrostatic force on a position of the commonmass.

A further understanding of the nature and the advantages of particularembodiments disclosed herein may be realized by reference of theremaining portions of the specification and the attached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1a shows a conceptual block diagram of a MEMS capacitive sensinginterface 11, in accordance with an embodiment of the invention.

FIG. 1b shows a conceptual block diagram of a MEMS capacitive sensinginterface 21, in accordance with another embodiment of the invention.

FIG. 1c shows a conceptual block diagram of a MEMS capacitive sensinginterface 31, in accordance with another embodiment of the invention.

FIG. 1d shows a conceptual block diagram of a MEMS capacitive sensinginterface 67, in accordance with another embodiment of the invention.

FIG. 2 shows a conceptual block diagram of a dual-mode MEMS system 71using a sensing technique for tuning, in accordance with an embodimentand method of the invention.

FIG. 3 shows a conceptual block diagram of a MEMS capacitive sensinginterface 103, in accordance with an embodiment of the invention.

FIG. 4 shows a conceptual block diagram of the interface 103 with anadded capacitor 94.

DETAILED DESCRIPTION OF EMBODIMENTS

The following describes a MEMS capacitive sensing interface used tomeasure the displacement of a MEMS device while reducing the effect ofelectrostatic spring softening. In one embodiment of the invention, theMEMS capacitive sensing interface has a feedback loop comprising a sensecapacitor, a feedback capacitor, and a capacitance measurement circuit,all of which are used to substantially eliminate dependence of the netelectrostatic force on the position of the MEMS device. In otherembodiments and methods of the invention, a resonance frequency of theMEMS device is tuned, allowing for tuning of the mechanical resonancemodes.

Referring now to FIG. 1a , a conceptual block diagram of a MEMScapacitive sensing interface 11 is shown, in accordance with anembodiment of the invention. The interface 11 is shown to include a MEMSdevice modeled as a common mass (m) 10, a spring (k_(x)) 20, and adamper (b) 30. For example, this model may represent one of anaccelerometer, gyroscope, compass, microphone or other MEMS device knownin the art. The damper in the model represents the effect of variousenergy loss mechanisms (such as air damping) on the behavior of the MEMSdevice. The interface 11 is further shown to include a sense capacitor(C_(s)(x)) 50, a feedback capacitor (C_(f)(x)) 81, and a capacitancemeasurement circuit comprising an amplifier 60, a feedback capacitor(C_(fb)) 61, and a sign inverter 91. A feedback loop comprises thecapacitive measurement circuit 60, the sense capacitor 50, and thefeedback capacitor 61.

The common mass 10 is shown to be flexibly attached to a fixed structure1 through the spring 20 and the damper 30. A force 17 is applied to thecommon mass 10. The common mass, 10, is shown to have two terminals, onesuch terminal is shown connected the common mass through a terminal 41of the sense capacitor 50 and another such terminal is shown connectedto the common mass 10 through a terminal 43 of the capacitor 81.

Another terminal of the sense capacitor 50, namely terminal 45, is showncoupled to the amplifier 60 and the feedback capacitor 61. The feedbackcapacitor 61 is shown coupled at another terminal to the output of theamplifier 60, which is also shown coupled to the input of the inverter91. The output of the inverter 91 is shown coupled to the terminal 47that serves as one of the terminals of the capacitor 81.

In the MEMS interface 11, an input force F 17 is applied to the commonmass 10. The resulting displacement of the mass, x 40, is sensed bymeasuring a corresponding change in sense capacitance C_(s) 50. A highvoltage bias (V_(b)) 80 is applied to the common mass 10. In anexemplary embodiment, the voltage 80 is in the range of 25V. The use ofhigh voltage biasing improves the sensitivity of the MEMS in that moreoutput charge flows through capacitor C_(s) (50) in response to a givendisplacement 40. However, the use of high voltage bias 80 also producesa first electrostatic force in capacitor C_(s) 50 acting on the commonmass 10. This force is position dependent and acts like a negativespring constant for small displacements. This negative spring constantis undesirable in that it modifies the dynamics of the MEMS interface. Asecond capacitor, C_(f) 81, is also biased with high voltage bias 80 andtherefore has associated with it a second position-dependentelectrostatic force acting on the common-mass 10. However, drivingterminal 47 of capacitor C_(f) 81 with a signal proportional to themeasured change in position 40 modifies the second force such that theposition dependence of the net force (the sum of the first and secondforces) is substantially modified, reduced or eliminated. By this means,the dynamics of the MEMS interface may be optimized and the undesirablenegative spring constant may be substantially reduced or eliminated.

To achieve this benefit, terminal 47 is driven as follows. A feedbackcapacitor (C_(fb)) 61 is shown connected to the output of the amplifier60 at one of its terminals and at an opposite one of its terminals, tothe terminal 45. The amplifier 60 effectively converts the change incapacitance of the capacitor 50 to a change in voltage and provides thegenerated voltage to the inverter 91 and terminal 47 of the capacitor81. Since the change in capacitance of the capacitor 50 is approximatelyproportional to the change in position 40, the voltage applied toterminal 47 is also proportional to the change in position 40. Each ofthe capacitors 61, 50, and 81 is a parallel-plate type of capacitors,thus, terminals 41, 43, 45, and 47 are commonly referred to as “plates”.

The interface 11 of FIG. 1a , provides a feedback force to eliminate theposition dependence of the net force and, consequently, the springsoftening. Feedback capacitor (C_(f)) 81 having a terminal residing onthe common mass 10 is used for this purpose. When the feedback gain isset for an exact cancellation of the position dependent force, theproposed circuit eliminates the spring softening. This condition also isequivalent to a constant charge operation, since the charge removed fromthe sense capacitor (C_(s)) 50 is replaced by charge placed on thefeedback capacitor 81, resulting in zero change in the total charge onthe MEMS device.

In FIG. 1a , a conceptual model of the interface 11 is shown at 13 wherethe effect of electrostatic forces acting across capacitors C_(s) 50 andC_(f) 81 are represented by two equivalent electrostatic springs k_(v)70 and k_(q) 100. With reference to this model, the conditions forcancellation of the spring softening can be derived.

Spring k_(v) 70 represents the position-dependent electrostatic forcebetween the electrodes of capacitors C_(s) 50 and C_(f) 81, and itsvalue is given by the following relationship:

$\begin{matrix}{k_{v} = {{- \frac{^{2}\left( {C_{s} + C_{f}} \right)}{x^{2}}}{\frac{V_{b}^{2}}{2}.}}} & {{Eq}.\mspace{14mu} (6)}\end{matrix}$

Note that the spring constant, k_(v) 70, is negative, which isindicative of electrostatic spring softening. Spring kq 100 representsthe action of the feedback loop acting on the common mass 10 via thefeedback capacitor Cf 81, and its value is given by the followingrelationship:

$\begin{matrix}{k_{q} = {\frac{1}{C_{fb}}\frac{C_{s}}{x}\frac{C_{f}}{x}{V_{b}^{2}.}}} & {{Eq}.\mspace{14mu} (7)}\end{matrix}$

Eliminating the position dependent force requires the followingrelationship between the capacitors 80 and 50:

$\begin{matrix}{{C_{fb} = \frac{C_{s}C_{f}}{C_{s} + C_{f}}},} & {{Eq}.\mspace{14mu} (8)}\end{matrix}$

which also maintains a constant charge on the MEMS device. Thisparticular condition creates a k_(q) 100 that cancels k_(v) 70. Thus,the interface 11 mitigates electrostatic spring softening inparallel-plate sense capacitors, such as the capacitors 50 and 81. It isalso worth noting that the resonance frequency, f_(res), of the systemis now a function of k_(q) as well, and can be expressed as:

$\begin{matrix}{f_{res} = {\sqrt{\frac{k_{x} + k_{v} + k_{q}}{m}}.}} & {{Eq}.\mspace{14mu} (9)}\end{matrix}$

It should be noted that observing the relationship of Eq. (8) leads tosubstantial cancellation of the electrostatic spring softening, but itdoes not null the motion of the position common mass 10. Rather, itallows the position of the common mass 10 to vary normally in responseto the applied force 17 in a manner as if there were essentially noelectrostatic spring softening. This exemplary embodiment of theinvention is different from position-nulling techniques known in the artin which a feedback network applies a force substantially equal to theapplied force 17. In particular, known position-nulling techniques donot cancel spring softening. Furthermore, the foregoing embodiment ofthe invention does not null changes in position of the common mass 10.Position-nulling techniques balance the total applied force acting onthe common mass 10, whereas the this embodiment eliminates dependence ofthe total force on position of the common mass 10. Accordingly, variousdisadvantages of position nulling (such as a need for high forcetransduction to null the total force and inverse dependence of theoutput signal of the amplifier 60 on the bias voltage, V_(b) 80) areadvantageously avoided by the various embodiments of the invention. Itis also noted that position-nulling is not generally operable to providefine tuning of the resonance frequency of the system, as would beindicated for the embodiment represented by Eq. (9). Rather,position-nulling is operable to significantly broaden the bandwidth of aMEMS system, in contrast to the various embodiments of the inventionoperable to fine tune the resonance frequency. Thus, the known techniqueof position-nulling by force feedback and the embodiments of theinvention serve different purposes and have different functionality.

In alternative embodiments, the interface 11 can also be applied to nonparallel-plate MEMS capacitors. For instance, a comb-based MEMScapacitor does not have spring softening. Thus, k_(v) 70 is equal to 0and kq 100 may be selected based on C_(fb) 61 to provide a desiredspring stiffness.

FIG. 1b shows a conceptual block diagram of a MEMS capacitive sensinginterface 21, in accordance with another embodiment of the invention. Aconceptual model of the interface 21 is shown at 23 and is analogous tothat which is shown at 13, in FIG. 1 a.

The interface 21 is analogous to the interface 11 except that thecapacitors 50 and 81 are shown to be connected to a common terminal 51,which is also shown connected to the common mass 10. That is, the senseand feedback capacitors, C_(s)(x) 50 and C_(f)(x) 80 are implemented bysplitting only the static electrode of the MEMS capacitor, as shown inFIG. 1 b.

In the embodiments of FIGS. 1a and 1b , the electrostatic springsoftening effect is reduced by measuring the change in the position 40of the MEMS device, while the MEMS device is subjected to aposition-dependent force associated with capacitors 50 and 81 andapplying an additional force to the common mass 10 via capacitor 81 withthe additional force being proportional to the measured change inposition 40. The constant of proportionality is selected so that the netforce has reduced dependence on the change in the position 40. In someembodiments, the two foregoing forces act in the same direction on thecommon mass 10 and in other embodiments, they act in opposite directionson the common mass 10.

It is further possible further possible to have another embodiment wherethe sense and feedback capacitors are placed on opposite sides of theMEMS device, an example of which is shown in FIG. 1c . FIG. 1c shows aconceptual block diagram of a MEMS capacitive sensing interface 31, inaccordance with another embodiment of the invention. A conceptual modelof the interface 31 is shown at 41.

The interface 31 is analogous to that of FIGS. 1a and 1b except that theinterface 31 lacks the inverter 91 and rather connects the output of theamplifier 35, which is analogous to the amplifier 60, to one of theterminals of the feedback capacitor 33 with an opposite terminal of thefeedback capacitor 33 being connected to the common mass 10. The sensecapacitor 37 is analogous to the capacitor 50 and the capacitor 39 isanalogous to the capacitor 61.

The embodiment of FIG. 1c does not require the capacitors 37 and 33 to ahave a common terminal. Such an implementation, however, will havemultiple electrical nodes within the MEMS devices, one associated witheach of capacitors 37 and 33. When sense and feedback capacitors areplaced on opposite sides of the MEMS device, feedback loop no longerpreserves constant charge on the MEMS device, even though it removesposition dependence of the electrostatic force.

FIG. 1d shows a conceptual block diagram of a MEMS capacitive sensinginterface 67, in accordance with another embodiment of the invention. Aconceptual model of the interface 67 is shown at 51, in FIG. 1d . Theinterface 67 is analogous to the interface 21 except that it includes afeedback circuit 96 connected between the output of the amplifier 83 andone of the terminals of the capacitor 37, and a switches 43 and 44 areshown at the same terminal of the capacitor 37, with switch 43 coupledto the amplifier 83 and switch 44 coupled to the feedback circuit 96.The amplifier 83 is otherwise analogous to the amplifier 60.

In the interface 67, sense and feedback operations are separated by timedivision multiplexing. Such an implementation allows using the samesense capacitor, i.e. capacitor 37, for both position sensing andfeedback. The sense circuit, made of the amplifier 83 and the capacitor81 which is analogous to the capacitor 61, measures the value ofcapacitance in one phase φ₁ 101, while the feedback circuit 96 appliesfeedback in the other phase φ₂ 102.

The embodiments of FIGS. 1a-1d can introduce electrostatic springstiffening via k_(q) which can be used to offset the electrostaticspring softening of k_(v). Introducing additional electrostatic springsoftening via k_(q) would also be possible by removing the inverter 91,shown in FIG. 1a , from the feedback loop. In alternative embodiments,such as shown in FIG. 2 and discussed below, both stiffening andsoftening allows the MEMS capacitance sensing interfaces of suchembodiments to advantageously tune a resonance frequency of the MEMSdevice. Since this feature allows tuning of the mechanical resonancemodes, it will be referred to as the “mode tuning” feature.

FIG. 2 shows a conceptual block diagram of an exemplary dual-mode MEMSsystem using a sensing technique for mode tuning, in accordance with anembodiment and method of the invention. In FIG. 2, a dual-mode MEMSsystem 71 is shown to include a spring 63, k_(d), a damper 65, attachedto the fixed structure 1 and to a mass 99 (m_(d)). The common mass 10 isshown to also be connected to a spring 67, k_(c), which is shownconnected to the mass 99. Common mass 10 also connects to spring 69,k_(s), and to the damper 71, b_(s), and further connected to thecapacitor 37 and a feedback capacitor 83. The capacitor 83 is shown alsoconnected to the output of a feedback gain k_(f) 92, which receives itsinput from the capacitor 81 and the output of the amplifier 83. Thecapacitor 37 is connected to the capacitor 81 and the amplifier 83 muchin the same manner as shown and discussed relative to FIG. 1d . In fact,the mass 10, spring 69, damper 71, capacitors 37, 83, 81, and amplifier83 are analogous to those of their counterparts in FIG. 1d . The dampersin the model represent the effects of various energy loss mechanisms(such as air damping) on the behavior of the dual-mode MEMS device.

The common mass 10 is operated upon much in the same way as discussedhereinabove relative to FIG. 1a except that the resonance frequency ofthe common mass 10 is changed by changing the gain 92, which allows forthis adjustable gain to better reduce and effectively cancel springsoftening upon the common mass 10. Changing the gain 92 allowsadjustment of the k_(q) 100 term higher or lower than the inherentspring softening k_(v) 70. By this means, the amplitude of the netelectrostatic force is adjusted.

Mode tuning is particularly useful when the sense system resonancefrequency is required to match a specific frequency, e.g. the mechanicalresonance frequency of a drive system. As an example, in FIG. 2, thedual-mode transducer or MEMS system uses the transcapacitance amplifier83 to tune the resonance frequency of the sense system to match thedrive frequency by removing the spring softening and also compensatingfor manufacturing mismatch between the two. Tunability of k_(q) 100 canbe introduced, for example, by adjusting the gain k_(f) 92. Such animplementation also has the advantage of decoupling frequency tuningfrom the transcapacitance gain of the sense circuit.

FIG. 3 shows a conceptual block diagram of a MEMS capacitive sensinginterface 103, in accordance with an embodiment of the invention. Alsoshown in FIG. 3 is a conceptual model of the interface 103, at 101. Theinterface 93 is analogous to the interface 11 except that the inverter91 is replaced with the signal conditioning circuit 110 having theeffect of introducing a derivative term to the feedback. The use of aderivative term in the signal conditioning circuit 110 acts to modifythe phase of the net electrostatic force. When the feedback loop of theembodiment of FIG. 3 is tuned for spring stiffening, the derivative termintroduces additional damping b_(q) 120 to MEMS dynamics, as follows:

F=m·s ² +b·s+k _(x)+(1+d·s)k _(q) =m·s ²+(b+b _(q))s+(k _(x) +k_(q)).  Eq. (10)

Since b_(q) 120 is introduced by active electronics, its noisecontribution is not directly related to the damping b_(q) 120. Thus,compared to the dissipative means, i.e. air damping or resistivedamping, active damping introduced by the proposed circuit can controlthe quality factor of the MEMS dynamics without a significant noisepenalty. Such introduction of damping can be used, for example, toimprove settling time of the MEMS system without sacrificing the noiseperformance.

The interface 103 electronics include MEMS dynamics directly in itsfeedback path, thus, critical circuit parameters such as noise andstability directly depend on MEMS dynamics. The electronic noisegenerated in the capacitance sensing circuit is expected to be asignificant noise source and can be modeled as a voltage source v. 93 atthe input of the amplifier circuit as shown in FIG. 4. FIG. 4 shows aconceptual block diagram of the interface 103 but with an addedparasitic capacitor 94. FIG. 4 further shows a graph of magnituderesponse vs. frequency of the noise- and signal-transfer functionsassociated with the interface of FIG. 4. The graph of FIG. 4 helps toshow how the presence of MEMS dynamics affects the noise transferfunction (NTF), at 141 to the output of the capacitance sensing circuit.Using small-signal force-balance equations for MEMS dynamics, it can beshown that the noise transfer function 141 of the system shown in FIG. 4is, as follows:

$\begin{matrix}{{{NTF} = {\frac{C_{p} + C_{s} + C_{fb}}{C_{fb}}\frac{{m \cdot s^{2}} + {b \cdot s} + k_{x} + k_{v} + {\frac{C_{s}^{\prime}}{C_{f}^{\prime}}\frac{C_{fb}}{C_{p} + C_{s} + C_{fb}}k_{q}}}{{m \cdot s^{2}} + {b \cdot s} + k_{x} + k_{v} + k_{q}}}},} & {{Eq}.\mspace{14mu} (11)}\end{matrix}$

where C_(s)′ and C_(f)′ represent first derivative of these capacitanceswith respect to the MEMS position.Furthermore, the signal transfer function (STF) 142 can be written as afunction of the transducer sensitivity (S_(c2v)) at the capacitancesensing circuit output and normalized MEMS dynamics:

$\begin{matrix}{{STF} = {S_{c\; 2v}\frac{k_{x} + k_{v} + k_{q}}{{m \cdot s^{2}} + {b \cdot s} + k_{x} + k_{v} + k_{q}}}} & {{Eq}.\mspace{14mu} (12)}\end{matrix}$

As seen in FIG. 4, both noise and signal are amplified by the mechanicalresonance controlled by the constant charge sensing scheme. In otherwords, both STF and NTF have a peak around the resonance frequencyf_(res) 143 set by the feedback loop 90. The noise transfer function,however, also has an additional notch 144 at approximately the resonancefrequency of MEMS without the feedback loop 90. In cases where thedesired sense signal does not have to be at the exact sense resonancefrequency f_(res) 143, the noise notch 144 introduced by the feedbackloop 90 can be exploited to significantly reduce the electronic noisecontribution.

Having MEMS dynamics in the feedback loop of the interface electronicsalso constitute additional circuit requirements for stability. Both thetranscapacitance feedback and charge replenishing feedback loops can bebroken at the output of the amplifier 95. Loop gain analysis performedaround the breaking point shows that MEMS dynamics create an additionalGain Margin requirement for the transcapacitance amplifier. As shown bythe expected loop gain as follows:

$\begin{matrix}{{{T(s)} = {{A(s)}\frac{{m \cdot s^{2}} + {b \cdot s} + k_{x} + k_{v} + k_{q}}{{m \cdot s^{2}} + {b \cdot s} + k_{x} + k_{v} + {\frac{C_{s}^{\prime}}{C_{f}^{\prime}}\frac{C_{fb}}{C_{p} + C_{s} + C_{fb}}k_{q}}}}},} & {{Eq}.\mspace{14mu} (13)}\end{matrix}$

where A(s) represents the open loop gain of the amplifier by itself,zeros of the loop gain are set by the dynamics under the influence ofthe constant charge sensing operation. Assuming the amplifiercharacteristics can be approximated by an integrator, complex zeros ofthe loop gain introduce the necessary phase shift for instability. Thus,having adequate gain margin (GM) around the resonance frequency f_(res)is required for stability of the proposed circuit.

In accordance with the foregoing, electrostatic spring stiffening isprovided thereby compensating for spring softening by using a second setof feedback capacitors to replenish the charge removed for sensing.Further, tuning of a resonance frequency of the MEMS device isperformed, in accordance with various methods and embodiments of theinvention.

The foregoing embodiments have been described in reference tosingle-ended circuit diagrams for the sake of clarity. It will beevident to one of ordinary skill that single-ended or differentialembodiments are possible within the scope and spirit of the invention.The foregoing embodiments have also been described in reference tocontinuous-time circuit diagrams. It will be evident to one of ordinaryskill that discrete-time embodiments are also possible within the scopeand spirit of the invention.

Thus, while particular embodiments have been described herein, latitudesof modification, various changes, and substitutions are intended in theforegoing disclosures, and it will be appreciated that in some instancessome features of particular embodiments will be employed without acorresponding use of other features without departing from the scope andspirit as set forth. Therefore, many modifications may be made to adapta particular situation or material to the essential scope and spirit.

What we claim is:
 1. A method of reducing electrostatic springsoftening, comprising: measuring a change in position of a masssubjected to a first force depending on the change in position, applyinga second force to the mass, the second force comprising a forceproportional to the measured change in position, the constant ofproportionality being selected so that the net force of the first andsecond forces has reduced dependence on the change in position.
 2. Themethod of claim 1, wherein the dependence on the change is position issubstantially eliminated.
 3. The method of claim 1, wherein the firstand second forces act in the same direction on the mass.
 4. The methodof claim 1, wherein the first and second forces act in oppositedirections on the mass.
 5. The method of claim 1, further comprising thestep of applying a high-voltage bias to the mass.
 6. The method of claim1, wherein the steps of measuring a change in position and applying asecond force are time division multiplexed.
 7. A method of tuning a MEMScapacitive sensing interface, comprising: measuring a change in positionof a mass of a MEMS device, applying a first force to the mass, thefirst force comprising a force proportional to the change in position,the constant of proportionality providing at least one of a phase shiftor amplitude shift in relation to the measured change in position, theconstant of proportionality being selected to adjust a dynamic responseof the MEMS device.
 8. The method of claim 7, wherein the constant ofproportionality provides an amplitude shift and the dynamic response isadjusted to tune a resonant frequency.
 9. The method of claim 7, whereinthe constant of proportionality provides a phase shift and the dynamicresponse is adjusted to tune a damping.
 10. The method of claim 7,further comprising the step of applying a high-voltage bias to the mass.11. The method of claim 7, wherein the steps of measuring a change inposition and applying a second force are time division multiplexed.